Abstract

The trigonal curve associated with the Vakhnenko equation is introduced by using the Lax matrix for the nth stationary positive flow. Based on the theory of the trigonal curve and the properties of the three kinds of Abel differentials, the Riemann theta function representation for the Baker-Akhiezer function is derived, from which the straightening out of flows in the Jacobian variety is exactly given through the Abel maps. We finally arrive at quasi-periodic solutions of the Vakhnenko equation.